An Engel condition with two generalized derivations in prime rings

Abstract

Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that for all where are fixed positive integers. Then, one of the following conditions holds: there exist such that and for all m = s, n = t and there exist and such that g(x) = xa and for all C is a finite field, the matrix ring over C for some integer and there exist such that g(x) = xa and h(x) = ux for all